Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 6 - Applications of the Integral - 6.4 The Method of Cylindrical Shells - Exercises - Page 312: 46


$\dfrac{408 \pi}{5} $

Work Step by Step

The volume of a revolution can be calculated as: $V=\pi \int_{m}^{n} (R^2_{outer}-r^2_{inner}) \ dy$ where, $R_{outer}=8-0=8$ and $ R_{inner}=8-(x^2+2)=6-x^2$ Now, $V=\pi \int_0^2 [(8)^2-(6-x^2)^2] \ dx \\=\pi \int_0^2 (64-36+12x^2-x^4) \ dx \\=\pi \int_0^2 (28+12x^2-x^4) \ dx \\= \pi (28 x+4x^3-\dfrac{x^5}{5}]_0^2 \\=\dfrac{408 \pi}{5} $
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